3. Prove the following statements: (a) If n is an even positive integer then n3 is an even positive integer. (b) If n3 is an odd positive integer and n is a positive integer then n is an odd positive integer.

Define an integer k to be odd if k − 1 is even. Write up a formal proof of the following, using an indirect proof: Claim: For any natural numbers m and n, if m is odd and n is odd, then m + n is even.

- A Pythagorean prime number is a prime number of the form 4n + 1, where n ≥ 1. A Mersenne prime number is a prime number of the form 2" – 1. A perfect number is a number that is equal to the sum of all of its divisors, including 1 but excluding the number itself. Consider the following statements: (1) Let m is the smallest Pythagorean prime number. Then, 2¹ – 1 is a Mersenne prime number. (2) If m is a perfect number, then 2 – 1 is a Mersenne prime number. Select one of the following choices: (a) (1) is True and (2) is False. (c) (1) and (2) are True. (b) (1) is False and (2) is True. (d) (1) and (2) are False.

12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2 = 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+ 2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.